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Theorem elab6g 3621
Description: Membership in a class abstraction. Class version of sb6 2088. (Contributed by SN, 5-Oct-2024.)
Assertion
Ref Expression
elab6g (𝐴𝑉 → (𝐴 ∈ {𝑥𝜑} ↔ ∀𝑥(𝑥 = 𝐴𝜑)))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem elab6g
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eleq1 2825 . 2 (𝑦 = 𝐴 → (𝑦 ∈ {𝑥𝜑} ↔ 𝐴 ∈ {𝑥𝜑}))
2 eqeq2 2748 . . . 4 (𝑦 = 𝐴 → (𝑥 = 𝑦𝑥 = 𝐴))
32imbi1d 341 . . 3 (𝑦 = 𝐴 → ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝐴𝜑)))
43albidv 1923 . 2 (𝑦 = 𝐴 → (∀𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝐴𝜑)))
5 df-clab 2714 . . 3 (𝑦 ∈ {𝑥𝜑} ↔ [𝑦 / 𝑥]𝜑)
6 sb6 2088 . . 3 ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))
75, 6bitri 274 . 2 (𝑦 ∈ {𝑥𝜑} ↔ ∀𝑥(𝑥 = 𝑦𝜑))
81, 4, 7vtoclbg 3528 1 (𝐴𝑉 → (𝐴 ∈ {𝑥𝜑} ↔ ∀𝑥(𝑥 = 𝐴𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1539   = wceq 1541  [wsb 2067  wcel 2106  {cab 2713
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814
This theorem is referenced by:  elabd2  3622  elabgt  3624  elabg  3628  sbc6g  3769  upwordnul  45109  upwordsing  45113
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